F X †’ 0 ∞ by F X Inf d X a a ˆˆ a Show F is Continuous
I tracked down the Doubtnut video on YouTube from which your graph comes. (The presentation is not entirely in English, but the math comes through well enough.) It should be said that the polynomial being discussed there is $ \ x^3 - \mathbf{3}x + 1 \ \ , $ so I'll describe what is being done to solve that problem and then adapt the method to the cubic polynomial under discussion on this page.
As is described in some of the other posts, to ask for the values of $ \ x \ $ at which $ \ f( \ f(x) \ ) \ = \ 0 \ \ $ is to seek those values (such as may exist) at which $ \ f(x) \ $ is equal to any of the zeroes of $ \ f(x) \ \ . $ This is to say that if $ \ r \ $ is a zero of the function, so that $ \ f(r) \ = \ 0 \ \ , $ then the value, say, $ \ x = a \ \ , $ for which $ \ f(a) \ = \ r \ \ $ is one for which $ \ f( \ f(a) \ ) \ = \ f(r) \ = \ 0 \ \ . $
Generally for most functions, solving analytically for the zeroes of a composition of two functions, or a self-composition in this problem, can be very difficult to impossible; we would usually find ourselves using a computational aid. As we are only asked to find the number of real zeroes of $ \ f( \ f(x) \ ) \ \ , $ we can resort to a graphical technique; but for that, we will need a graph of $ \ f(x) \ \ . $
What is being done in about the first half of the video is to construct this necessary graph. (The fact that this is being done, rather than simply getting a plot from a computer utility or on-line source, leads me to suspect that this is a "contest-math" problem, so no device would be available.) The first derivative equation $ \ f'(x) \ = \ 3x^2 - 3 \ = \ 0 \ \ $ is used to find the local extrema ("turning-points") of the function curve, $ \ x = \pm 1 \ \ , $ and the function is then evaluated to obtain $ \ f(-1) \ = \ 3 \ $ and $ \ f(1) \ = \ -1 \ \ . $ From our familiarity with graphs of polynomials (and cubics in particular) and a little help from the Intermediate Value Theorem, the presenter then sketches a version of the following.
This graph is used to obtain the number and approximate values of the zeroes of $ \ f(x) \ \ . $ It is observed that there are three zeroes: $ \ a \ \ , $ "somewhere between -1 and -2" , $ \ b \ \ , $ "between 0 and 1", and $ \ c \ \ $ "somewhere between 1 and 2 ". (This approach usually does not require high precision location of the zeroes.)
The presenter then draws in three "level-lines" at $ \ y = a \ , y = b \ , \ $ and $ \ y = c \ \ , $ and then looks at the intersections of these lines with the graph of $ \ f(x) \ \ . $ We don't really care what the $ \ x-$ coordinates of those intersection points are: we just want to count them. We find that there is one value of $ \ x \ $ ($ \ \approx -2.1 \ \ ; $ call it $ \ x_1 \ $ for the moment) at which $ \ f(x) = a \ \ , $ so $ \ f( \ f(x_1) \ ) \ = \ f(a) \ = \ 0 \ \ . $ There are three intersection points for $ \ f(x) \ = \ b \ $ and three more for $ \ f(x) \ = \ c \ \ , $ so we may conclude that there are seven real values of $ \ x \ $ for which $ \ f( \ f(x) \ ) \ = \ 0 \ \ $ (and it is clear from the graph that there are all distinct). [The presenter apparently forgets to deal with $ \ f(x) \ = \ c \ \ $ and so comes up with a total of four, which is remarked on in the video comments.]
If we apply this procedure to $ \ x^3 - x + 1 \ \ , $ the first derivative equation becomes $ \ f'(x) \ = \ 3x^2 - 1 \ = \ 0 \ \ , $ indicating that the local extremal values are $ \ f \left(-\frac{1}{\sqrt3} \right) \ = \ 1 + \ \frac{2}{3\sqrt3} \ \ $ and $ \ f \left(\frac{1}{\sqrt3} \right) \ = \ 1 - \ \frac{2}{3\sqrt3} \ > 0 \ \ . $ [This is noted in a number of the other posted answers.] For the graphical analysis, this means that the curve for $ \ f(x) \ $ intersects the $ \ x-$axis just once "somewhere between -1 and -2 " . Drawing in the appropriate level-line this time produces only one intersection with the function curve, so there is just one real value of $ \ x \ \approx \ -1.6 \ \ $ here for which $ \ f( \ f(x) \ ) \ = \ 0 \ \ . $
Source: https://math.stackexchange.com/questions/4185766/what-does-ffx-0-mean